{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T08:53:40Z","timestamp":1772528020945,"version":"3.50.1"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":4519,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[2001,9]]},"abstract":"<jats:p><jats:bold>\u00a7 1. Introduction<\/jats:bold>. In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A <jats:italic>Polish metric space<\/jats:italic> is a complete separable metric space (<jats:italic>X, d<\/jats:italic>).<\/jats:p><jats:p>Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks (in the beginning of Section 2): \u201cThe classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not \u2018smooth\u2019 in the modern terminology.\u201d Our Theorem 2.1 below quantifies precisely the enormity of this task.<\/jats:p><jats:p>After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier.<\/jats:p><jats:p>The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.<\/jats:p><jats:p>The rest of this section is devoted to an introduction of some basic ideas of a theory of complexity for classification problems, which will help to put our results in perspective. Detailed expositions of this general theory can be found, e.g., in Hjorth [2000], Kechris [1999], [2001].<\/jats:p>","DOI":"10.2307\/2687754","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:17:34Z","timestamp":1146986254000},"page":"361-375","source":"Crossref","is-referenced-by-count":20,"title":["Polish Metric Spaces: Their Classification and Isometry Groups"],"prefix":"10.1017","volume":"7","author":[{"given":"John D.","family":"Clemens","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Su","family":"Gao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander S.","family":"Kechris","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600005631_ref008","first-page":"323","volume-title":"General topology and its relations to modern analysis and algebra, VI (Prague, 1986)","volume":"16","author":"Kat\u011btov","year":"1988"},{"key":"S1079898600005631_ref006","volume-title":"Classification and orbit equivalence relations","volume":"75","author":"Htorth","year":"2000"},{"key":"S1079898600005631_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511735264"},{"key":"S1079898600005631_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF01411496"},{"key":"S1079898600005631_ref002","unstructured":"Clemens J. D. [2001], Ph.D. thesis , University of California, Berkeley."},{"key":"S1079898600005631_ref005","volume-title":"Metric structures for Riemannian and non-Riemannian spaces","volume":"152","author":"Gromov","year":"1999"},{"key":"S1079898600005631_ref010","doi-asserted-by":"crossref","unstructured":"Kechris A. S. [1999], New directions in descriptive set theory, this Bulletin, vol. 5, no. 2, pp. 161\u2013179.","DOI":"10.2307\/421088"},{"key":"S1079898600005631_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF01077284"},{"key":"S1079898600005631_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1994-1149121-0"},{"key":"S1079898600005631_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"S1079898600005631_ref004","unstructured":"Gao S. and Kechris A. S. [2000], On the classification of Polish metric spaces up to isom-etry, preprint."},{"key":"S1079898600005631_ref018","first-page":"181","article-title":"On the group of isometries of the Urysohn universal metric space","volume":"31","author":"Uspenski\u01d0","year":"1990","journal-title":"Commentationes Mathematicae Universitatis Carolinae"},{"key":"S1079898600005631_ref019","first-page":"374","article-title":"\u00dcber metrisch homogene B\u00e4ume","volume":"6","author":"van Dantzig","year":"1928","journal-title":"Abhandlungen aus dem Mathematischen Seminar der Universit\u00e4t Hamburg"},{"key":"S1079898600005631_ref016","first-page":"74","article-title":"Sur en espace m\u00e9trique universel","volume":"51","author":"Urysohn","year":"1927","journal-title":"Bulletin des Sciences Math\u00e9matiques"},{"key":"S1079898600005631_ref013","volume-title":"Banach spaces of continuous functions","author":"Semadeni","year":"1971"},{"key":"S1079898600005631_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0085620"},{"key":"S1079898600005631_ref020","doi-asserted-by":"publisher","DOI":"10.1070\/RM1998v053n05ABEH000069"},{"key":"S1079898600005631_ref012","unstructured":"Manoussos A. and Strantzalos P. 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